300 HELL SYSTEM TECHNICAL JOURNAL 



of Sequence 1 is shown in Fig. 4. Each rectangle represents a wave- 

 filter of ladder type having the two mid-point image impedances 

 as indicated. The operation symbol between each succeeding pair of 

 rectangles shows what operation has been performed and the arrow 

 points towards the derived structure of higher order, being placed in 

 line with the image impedances which are identical for the pair. Thus 

 it is seen that each derived structure has one identical and one more 

 general image impedance than the preceding structure. In the 

 sequence the new image impedances appear alternately at mid-series 

 and mid-shunt points, beginning with the latter here. 



The series and shunt impedances of the different structures which 

 become more and more complicated with increase in parameters are 

 derived by performing the above operations but their detailed con- 

 sideration will be deferred to a later point. 



The transfer constants of the various members of this sequence are 

 found by carrying out the proper operations based upon formulas (3), 

 (5) and (6) and can be expressed by one formula, namely 



cosh 7,(g) = 1 + YTir^^TKfATTi^ ' ^ ^ 



where g = 1, w, mm', mm'm", etc., in a decreasing sequence.^" The 

 value of g for the structure of any order is equal to the product of all 

 of its parameters, the first value above, g = 1, being that of the 

 "constant k'' wave-filter. This is, for example, because by (3) 



^ 1 + ( 1 - m-m m" ) ( Uk+i Vk) 



The image impedances in Sequence 1 which are derived in a corre- 

 sponding manner have these formulas. 



W.2k{m) = T/ro,[l+a(f/;t+iF,)], 



^'^^■("'' "' ) ^ \:i+a'im+m)T ' ^^^'^ 



, ,. W,ll+a(U,+iV,)T^+a"(U,+ iV,):\ 

 IT .A.(m, ni , m ) ^ [l+a'(t/.+ nO,.)] 



i» Computations for the transfer constant can be made accurately from formulas 

 for cosh-i (x + iv) given in Appendix III of the paper " Distortion Correction in 

 Electrical Circuits with Constant Resistance Recurrent Networks, O. J. Zobel, 



B. S. T. /., July, 1928. 



